This page will document useful transformations of elliptic integrals and relations among them, along with their derivations. These can be applied to simplifications of analytic formulae or numerical evaluations.
Definitions of Integrals
Complete Integrals of Negative Parameter
Reciprocal Modulus Transformations
Elliptic Nome on the Real Axis
The incomplete elliptic integrals of the first, second and third kinds in trigonometric form are
with in these defining integrals. The elliptic parameter m will be used throughout this presentation rather than the traditional elliptic modulus k for two reasons:
1) The resulting formulae are generally cleaner and simpler.
2) Computer algebra systems generally use the parameter rather than the modulus. Having formulae in the same form reduces unintended errors in application.
The elliptic parameter and modulus are related by . There is also traditionally a complementary elliptic modulus defined by , and functions of this complementary modulus are often denoted with a prime.
While there is a variety of mathematical notation for the elliptic integrals, it appears to be common to distinguish those using the parameter with a vertical bar before that argument, while those using the modulus are denoted with a comma before that argument.
The transformation gives the incomplete elliptic integrals in Legendre normal form:
The angular variable is left as an argument on the left-hand sides to aid in avoiding errors in application.
Setting the angular argument equal to defines the complete elliptic integrals:
A simple angular substitution of the form
interchanges sines and cosines while reversing the order of integration. This implies that the complete integrals can be equally defined using sines or cosines, since the integrals are otherwise identical after this substitution.
That fact can be used to evaluate the complete integrals for negative parameter quite easily:
With an argument of the denominator of the complete integral of the first kind acquires a zero. Since the real part of the inverse sine function is always less than , split the integral into two parts:
These two integrals can be converted to complete elliptic integrals with trigonometric substitutions that preserve the valid endpoint while scaling the other to a value appropriate for a complete integral. For the integral in the real part, the substitution that preserves the lower endpoint is
under which the integral becomes
For the integral in the imaginary part, the substitution that preserves the upper endpoint is
under which the integral becomes
since the complete integrals can be defined with either sines or cosines as noted above. The reciprocal modulus transformation for the complete elliptic integral of the first kind is thus
This result technically only holds for : for the imaginary part of the result requires a change in sign.
For this relation can be used to separate the real and imaginary parts of the complete integral when written in the form
since then the arguments on the right-hand side are both within the defining range of convergence.
The elliptic nome is defined by
For the arguments of the elliptic integrals are within the defining range, so that both integrals as well as the nome are real.
For the arguments of both integrals are outside the defining range. The complete integral with negative parameter has been given above, and the integral with argument greater than one can be rewritten using the reciprocal modulus transformation. The ratio of the two integrals is
so that the elliptic nome for real negative argument is
and since both arguments are now within the defining range, the nome is purely real.
For the arguments of both integrals are again outside the defining range but with their respective roles reversed. The ratio of the two integrals is now
The arguments on the right-hand side are now within the defining range, so that this ratio is explicitly separated into real and imaginary parts. The nome itself
is no longer entirely real in this domain.
Uploaded 2017.11.20 — Updated 2020.04.02 analyticphysics.com